6. Given the graph below, do the following: a) Eulerian Cycles and Paths: Add an edge to the above that the graph is still simple but now has an Eulerian Cycle or an Eulerian Path. What edge was added? Justify your answer by finding the Eulerian Cycle or Eulerian Path, listing the vertices in order traversed. b) Hamiltonian Cycles and Paths: i. Using the original graph without the edge added in a), is it possible to determine if the graph has a Hamiltonian Cycle by using Dirac's Theorem? Justify your answer. Using the original graph without the edge added in a), is the graph Bipartite? Justify your answer. Note: If it is possible to do a 2-coloring as justification, show it on a copy of the graph; if it is not possible you should explain why that is the case. If the graph is Bipartite, is it Complete? Justify your answer. What can you conclude about the existence of a Hamiltonian Cycle? Hamiltonian Path? Justify your answer. If you still cannot conclude about the existence of a Hamiltonian Cycle or Hamiltonian Path, can you find either or both using trial-and-error? Justify your ansv answer. 11.